# Mutual fund separation theorem Mutual fund separation theorem In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, sob certas condições, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio. Here a mutual fund refers to any specified benchmark portfolio of the available assets. There are two advantages of having a mutual fund theorem. Primeiro, if the relevant conditions are met, it may be easier (or lower in transactions costs) for an investor to purchase a smaller number of mutual funds than to purchase a larger number of assets individually. Segundo, from a theoretical and empirical standpoint, if it can be assumed that the relevant conditions are indeed satisfied, then implications for the functioning of asset markets can be derived and tested.

Conteúdo 1 Portfolio separation in mean-variance analysis 1.1 No risk-free asset 1.2 One risk-free asset 2 Portfolio separation without mean-variance analysis 3 References Portfolio separation in mean-variance analysis Portfolios can be analyzed in a mean-variance framework, with every investor holding the portfolio with the lowest possible return variance consistent with that investor's chosen level of expected return (called a minimum-variance portfolio), if the returns on the assets are jointly elliptically distributed, including the special case in which they are jointly normally distributed. Under mean-variance analysis, it can be shown that every minimum-variance portfolio given a particular expected return (isso é, every efficient portfolio) can be formed as a combination of any two efficient portfolios. If the investor's optimal portfolio has an expected return that is between the expected returns on two efficient benchmark portfolios, then that investor's portfolio can be characterized as consisting of positive quantities of the two benchmark portfolios.

No risk-free asset To see two-fund separation in a context in which no risk-free asset is available, using matrix algebra, deixar {displaystyle sigma ^{2}} be the variance of the portfolio return, deixar {mostre o estilo dele } be the level of expected return on the portfolio that portfolio return variance is to be minimized contingent upon, deixar {estilo de exibição r} be the vector of expected returns on the available assets, deixar {estilo de exibição X} be the vector of amounts to be placed in the available assets, deixar {estilo de exibição W.} be the amount of wealth that is to be allocated in the portfolio, e deixar {estilo de exibição 1} be a vector of ones. Then the problem of minimizing the portfolio return variance subject to a given level of expected portfolio return can be stated as Minimize {displaystyle sigma ^{2}} subject to {estilo de exibição X^{T}r=mu } e {estilo de exibição X^{T}1=W} where the superscript {displaystyle ^{T}} denotes the transpose of a matrix. The portfolio return variance in the objective function can be written as {displaystyle sigma ^{2}=X^{T}VX,} Onde {estilo de exibição V} is the positive definite covariance matrix of the individual assets' returns. The Lagrangian for this constrained optimization problem (whose second-order conditions can be shown to be satisfied) é {displaystyle L=X^{T}VX+2lambda (mu -X^{T}r)+2e (W-X^{T}1),} with Lagrange multipliers {lambda de estilo de exibição } e {estilo de exibição eta } . This can be solved for the optimal vector {estilo de exibição X} of asset quantities by equating to zero the derivatives with respect to {estilo de exibição X} , {lambda de estilo de exibição } , e {estilo de exibição eta } , provisionally solving the first-order condition for {estilo de exibição X} em termos de {lambda de estilo de exibição } e {estilo de exibição eta } , substituting into the other first-order conditions, solving for {lambda de estilo de exibição } e {estilo de exibição eta } in terms of the model parameters, and substituting back into the provisional solution for {estilo de exibição X} . The result is {estilo de exibição X^{matemática {opt} }={fratura {C}{Delta }}[(^{T}V^{-1}r)V^{-1}1-(1^{T}V^{-1}r)V^{-1}r]+{fratura {dentro }{Delta }}[(1^{T}V^{-1}1)V^{-1}r-(^{T}V^{-1}1)V^{-1}1]} Onde {displaystyle Delta =(^{T}V^{-1}r)(1^{T}V^{-1}1)-(^{T}V^{-1}1)^{2}>0.} For simplicity this can be written more compactly as {estilo de exibição X^{matemática {opt} }=alpha W+beta mu } Onde {alfa de estilo de exibição } e {beta de estilo de exibição } are parameter vectors based on the underlying model parameters. Now consider two benchmark efficient portfolios constructed at benchmark expected returns {mostre o estilo dele _{1}} e {mostre o estilo dele _{2}} and thus given by {estilo de exibição X_{1}^{matemática {opt} }=alpha W+beta mu _{1}} e {estilo de exibição X_{2}^{matemática {opt} }=alpha W+beta mu _{2}.} The optimal portfolio at arbitrary {mostre o estilo dele _{3}} can then be written as a weighted average of {estilo de exibição X_{1}^{matemática {opt} }} e {estilo de exibição X_{2}^{matemática {opt} }} do seguinte modo: {estilo de exibição X_{3}^{matemática {opt} }=alpha W+beta mu _{3}={fratura {dentro _{3}-dentro _{2}}{dentro _{1}-dentro _{2}}}X_{1}^{matemática {opt} }+{fratura {dentro _{1}-dentro _{3}}{dentro _{1}-dentro _{2}}}X_{2}^{matemática {opt} }.} This equation proves the two-fund separation theorem for mean-variance analysis. For a geometric interpretation, see the Markowitz bullet.

One risk-free asset If a risk-free asset is available, then again a two-fund separation theorem applies; but in this case one of the "funds" can be chosen to be a very simple fund containing only the risk-free asset, and the other fund can be chosen to be one which contains zero holdings of the risk-free asset. (With the risk-free asset referred to as "money", this form of the theorem is referred to as the monetary separation theorem.) Thus mean-variance efficient portfolios can be formed simply as a combination of holdings of the risk-free asset and holdings of a particular efficient fund that contains only risky assets. The derivation above does not apply, Contudo, since with a risk-free asset the above covariance matrix of all asset returns, {estilo de exibição V} , would have one row and one column of zeroes and thus would not be invertible. Em vez de, the problem can be set up as Minimize {displaystyle sigma ^{2}} subject to {estilo de exibição (W-X^{T}1)r_{f}+X^{T}r=mu ,} Onde {estilo de exibição r_{f}} is the known return on the risk-free asset, {estilo de exibição X} is now the vector of quantities to be held in the risky assets, e {estilo de exibição r} is the vector of expected returns on the risky assets. The left side of the last equation is the expected return on the portfolio, desde {estilo de exibição (W-X^{T}1)} is the quantity held in the risk-free asset, thus incorporating the asset adding-up constraint that in the earlier problem required the inclusion of a separate Lagrangian constraint. The objective function can be written as {displaystyle sigma ^{2}=X^{T}VX} , where now {estilo de exibição V} is the covariance matrix of the risky assets only. This optimization problem can be shown to yield the optimal vector of risky asset holdings {estilo de exibição X^{matemática {opt} }={fratura {(mu -Wr_{f})}{(r-1r_{f})^{T}V^{-1}(r-1r_{f})}}V^{-1}(r-1r_{f}).} Of course this equals a zero vector if {displaystyle mu =Wr_{f}} , the risk-free portfolio's return, in which case all wealth is held in the risk-free asset. It can be shown that the portfolio with exactly zero holdings of the risk-free asset occurs at {displaystyle mu ={tfrac {Wr^{T}V^{-1}(r-1r_{f})}{1^{T}V^{-1}(r-1r_{f})}}} and is given by {estilo de exibição X^{*}={fratura {C}{1^{T}V^{-1}(r-1r_{f})}}V^{-1}(r-1r_{f}).} It can also be shown (analogously to the demonstration in the above two-mutual-fund case) that every portfolio's risky asset vector (isso é, {estilo de exibição X^{matemática {opt} }} for every value of {mostre o estilo dele } ) can be formed as a weighted combination of the latter vector and the zero vector. For a geometric interpretation, see the efficient frontier with no risk-free asset.

Portfolio separation without mean-variance analysis If investors have hyperbolic absolute risk aversion (HARA) (including the power utility function, logarithmic function and the exponential utility function), separation theorems can be obtained without the use of mean-variance analysis. Por exemplo, David Cass and Joseph Stiglitz showed in 1970 that two-fund monetary separation applies if all investors have HARA utility with the same exponent as each other.: ch.4  More recently, in the dynamic portfolio optimization model of Çanakoğlu and Özekici, the investor's level of initial wealth (the distinguishing feature of investors) does not affect the optimal composition of the risky part of the portfolio. A similar result is given by Schmedders. References ^ Chamberlain, G (1983). "A characterization of the distributions that imply mean-variance utility functions". Revista de Teoria Econômica. 29: 185–201. doi:10.1016/0022-0531(83)90129-1. ^ Owen, J.; Rabinovitch, R. (1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice". Journal of Finance. 38 (3): 745–752. doi:10.1111/j.1540-6261.1983.tb02499.x. ^ Merton, Roberto; Setembro (1972). "An analytic derivation of the efficient portfolio frontier" (PDF). Journal of Financial and Quantitative Analysis. 7 (4): 1851–1872. doi:10.2307/2329621. HDL:1721.1/46832. JSTOR 2329621. ^ Cass, Davi; Stiglitz, Joseph (1970). "The structure of investor preferences and asset returns, and separability in portfolio allocation". Revista de Teoria Econômica. 2 (2): 122-160. doi:10.1016/0022-0531(70)90002-5. ^ Huang, Chi-fu, and Robert H. Litzenberger, Foundations for Financial Economics, Holanda do Norte, 1988. ^ Çanakoğlu, Ethem; Özekici, Süleyman (2010). "Portfolio selection in stochastic markets with HARA utility functions". European Journal of Operational Research. 201 (2): 520–536. doi:10.1016/j.ejor.2009.03.017. ^ Schmedders, Karl H. (Junho 15, 2006) "Two-fund separation in dynamic general equilibrium," SSRN Working Paper Series. https://ssrn.com/abstract=908587 Categories: Portfolio theoriesEconomics theoremsFinancial economics

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